Evaluation of Karhunen–Loève, Spectral, and Sampling Representations for Stochastic Processes
Publication: Journal of Engineering Mechanics
Volume 132, Issue 2
Abstract
The Karhunen–Loève, spectral, and sampling representations, referred to as the KL, SP, and SA representations, are defined and some features/limitations of KL-, SP-, and SA-based approximations commonly used in applications are stated. Three test applications are used to evaluate these approximate representations. The test applications include (1) models for non-Gaussian processes; (2) Monte Carlo algorithms for generating samples of Gaussian and non-Gaussian processes; and (3) approximate solutions for random vibration problems with deterministic and uncertain system parameters. Conditions are established for the convergence of the solutions of some random vibration problems corresponding to KL, SP, and SA approximate representations of the input to these problems. It is also shown that the KL and SP representations coincide for weakly stationary processes.
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References
Adler, R. J. (1981). The geometry of random fields, Wiley, New York.
Baroth, J., Bodé, L., and Fogli, M. (2003). “Numerical convergence of a spectral stochastic finite element method (SSEFEM) in lognormal context.” Proc., 9th Int. Conf. on Applications of Statistics and Probability in Civil Engineering, ICASP9, San Francisco, A. S. Der Kiureghian and J. P. Madanat, eds., Millpress, The Netherlands, 217–224.
Brabenec, R. L. (1990). Introduction to real analysis, PWS-KENT, Boston.
Cramer, H., and Leadbetter, M. R. (1967). Stationary and related stochastic processes, Wiley, New York.
Davenport, W. B., and Root, W. L. (1958). An introduction to the theory of random signals and noise, McGraw-Hill, New York.
Deodatis, G. (1996). “Non-stationary stochastic vector processes: Seismic ground motion applications.” Probab. Eng. Mech., 11, 149–168.
Field, R. V., Jr., and Grigoriu, M. (2004), “On the accuracy of the polynomial chaos approximation.” Probab. Eng. Mech., 19(1–2), 65–80.
Ghanem, R. G., and Spanos, P. D. (1991). Stochastic finite elements: A spectral approach, Springer, New York.
Gioffré, M., Gusella, V., and Grigoriu, M. (2000). “Simulation of non-Gaussian field applied to wind pressure fluctuations.” Probab. Eng. Mech., 5(4), 339–346.
Gohberg, I., and Goldberg, S. (1980). Basic operator theory, Birkhäuser, Boston.
Grigoriu, M. (1993). “On the spectral representation method in simulation.” Probab. Eng. Mech., 8(2), 75–90.
Grigoriu, M. (1995). Applied non-Gaussian processes: Examples, theory, simulation, linear random vibration, and MATLAB solutions, Prentice-Hall, Englewoods Cliffs, N.J.
Grigoriu, M. (2002). Stochastic calculus. Applications in science and engineering, Birkhäuser, Boston.
Grigoriu, M. (2004). “Spectral representation for a class of non-Gaussian processes.” J. Eng. Mech., 130(5), 541–546.
Grigoriu, M., Ditlevsen, O., and Arwade, S. R. (2003). “A Monte Carlo simulation model for staionary non-Gaussian processes.” Probab. Eng. Mech., 18(1), 87–95.
Hernández, D. B. (1995). Lectures on probability and second order random fields, World Scientific, London.
Kloeden, P. E., and Platen, E. (1992). Numerical solutions of stochastic differential equations, Springer, New York.
Mikosch, T. (1998). Elementary stochastic calculus, World Scientific, N.J.
Puig, B., Poirion, F., and Soize, C. (2002). “Non-Gaussian simulation using Hermite polynomials expansion: Convergence and algorithms.” Probab. Eng. Mech., 17(3), 253–264.
Resnick, S. I. (1998). A probability path, Birkhäuser, Boston.
Sakamoto, S., and Ghanem, R. (2002). “Simulation of multi-dimensional non-Gaussian non-stationary random fields.” Probab. Eng. Mech., 17(2), 167–176.
Shinozuka, M., and Deodatis, G. (1991). “Simulation of stochastic processes by spectral representation.” Appl. Mech. Rev., 44(4), 191–203.
Soong, T. T., and Grigoriu, M. (1993). Random vibration of mechanical and structural systems, Prentice-Hall, Englewood Cliffs, N.J.
Tolstov, G. P. (1962). Fourier series, Dover, New York.
Van Trees, H. L. (1968). Detection, estimation, and modulation theory, Vol. 1, Wiley, New York.
Wong, E., and Hajek, B. (1985). Stochastic processes in engineering systems, Springer, New York.
Information & Authors
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Published In
Journal of Engineering Mechanics
Volume 132 • Issue 2 • February 2006
Pages: 179 - 189
Copyright
© 2005 ASCE.
History
Received: May 5, 2004
Accepted: Apr 14, 2005
Published online: Feb 1, 2006
Published in print: Feb 2006
Notes
Note. Associate Editor: Gerhart I. Schueller
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